To address the aforementioned problems, we first describe  our proposed construction approach to find a tractable family of mixed-monotone remainder-form decomposition functions. Then, by characterizing their tightness, we can determine the tightest decomposition function among the proposed family. Further, we present a novel set inversion algorithm that serves as an alternative and improves on existing approaches, e.g., SIVIA in [1]}) and \(\mathcal {I}_G\)  in [2]}).
